Ind. Eng. Chem. Res. 1998, 37, 107-119
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Sequential Design Method for Multivariable Decoupling and Multiloop PID Controllers Shing-Jia Shiu and Shyh-Hong Hwang* Department of Chemical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, Republic of China
Underlying algorithms for designing multivariable decoupling and multiloop PI/PID controllers in a sequential fashion are addressed. A single-loop technique, composed of biased relay identification schemes and tuning formulae leading to the minimum weighted integral of square error, is developed to tune each loop in the predetermined sequence of loop closing. The proposed tuning technique is appropriate for a wide range of process dynamics in a multivariable environment. A method is then proposed to design decouplers to compensate for the effect of interactions and tune the resultant weakly interacting, single-loop PI/PID controllers sequentially. The decouplers, together with the single-loop controllers, constitute the multivariable decoupling controller. If the interactions are not significant, multiloop PI/PID controllers, which do not incorporate decouplers, could be employed. Simulation and comparative results are shown for one 2 × 2 and one 3 × 3 multivariable system from the literature. Despite its simplicity, the proposed design method yields superior multivariable designs on the basis of performance, robust stability, and integrity. 1. Introduction Any process capable of manufacturing or refining a product cannot operate satisfactorily within a single control loop. Virtually each unit operation requires at least two control loops to maintain the desired production rate and product quality (Shinskey, 1988). A large number of genuine multiloop control systems, which are made up of single-input/single-output (SISO) controllers acting in a multiloop fashion, have been reported (Vinante and Luyben, 1972; Wood and Berry, 1973; Ogunnaike and Ray, 1979; Tyreus, 1982). For such systems, loop interactions can arise and cause difficulties in feedback controller design. Cross-couplings of the process variables prevent the control engineer to design each loop independently. Adjusting controller parameters of one loop affects the performance of another, sometimes to the extent of destabilizing the entire system. To ensure stability, many industrial multiloop SISO controllers are tuned loosely, which causes inefficient operation and higher energy costs. There are multiloop design methods that treat the multiloop system as an entity. Niederlinski (1971) proposed a heuristic method based on a generalization of the classical single-loop tuning method of Ziegler and Nichols (1942) to tune multiloop PID controllers. The method has not gained wide acceptance because of its complexity and some reports of poor performance (Waller, 1984). Luyben (1986) proposed the biggest log modulus tuning (BLT) method for multiloop PI controllers. The method first tunes each individual PI controller following the single-loop Ziegler-Nichols rules and then detunes the entire system by a single factor to meet a specific stability requirement. The method of Basualdo and Marchetti (1990) is a modification of the BLT * Author to whom correspondence should be addressed. Telephone: 886-6-2757575 ext. 62661. Fax: 886-6-2344496. E-mail:
[email protected].
method. First, the individual controllers are designed independently based on the internal model control (IMC) structure (Garcia and Morari, 1982). Then, a single parameter is employed to adjust the multiloop system until robust stability and performance conditions are satisfied. The latter two methods have the disadvantage of requiring excessive modeling effort to seek a complete transfer function matrix. The idea of sequential design was used for multiloop control systems in recent years (O’Reilly and Leithead, 1991; Chiu and Arkun, 1992; Loh et al., 1993; Shen and Yu, 1994). According to specific sequential algorithms, the multivariable design problem is decomposed into a sequence of SISO design problems. Consequently, multiple single-loop designs can be employed by taking account of interactions in a sequential fashion. In this way, Loh et al. (1993) and Shen and Yu (1994) applied the single-loop relay technique of A° stro¨m and Ha¨gglund (1984) to design multiloop PI controllers. However, the conventional relay technique based on the describing function stipulates that the output response resemble a sinusoidal wave, which is often not the case in the identification of multiloop systems in a sequential fashion. There are also techniques for the design of true multivariable controllers that utilize all available process outputs jointly to make decisions on all inputs. With such controllers, it is possible to eliminate the effect of interactions between the process variables. Such techniques as the optimal control (LQ), the dynamic matrix control (DMC; Cutler and Ramaker, 1980), and the internal model control (IMC; Garcia and Morari, 1982) methods require the full knowledge of the process and the resultant controllers are often quite complex. Other approaches, such as Rosenbrock’s inverse Nyquist array (INA) method, make use of interaction compensators (decouplers) to eliminate the interactions between the
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108 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
loops. The design algorithms are computationally involved and therefore less appealing to practicing engineers. The objective of this work is to provide process engineers with an easy-to-use method to design multivariable decoupling and multiloop PI/PID controllers with very little prior knowledge about multivariable plants. The unknown multivariable plant is assumed square and open-loop stable. The design method must achieve good performance, robust stability, and integrity with minimal engineering effort. Underlying algorithms are first presented to tune multiple PI/PID control loops in a multivariable plant. A single-loop tuning technique, composed of biased relay identification schemes and tuning formulae, is applied to each loop in the predetermined sequence of loop closing. The tuning technique is versatile for a wide range of process dynamics prevailing in a multivariable environment. Unlike the conventional relay technique, the proposed identification schemes require only the existence of sustained oscillations and always lead to exact estimates of the frequency responses at zero and relay frequencies. If the multivariable system experiences severe interactions, a method is proposed to design decouplers to compensate for the effect of interactions. Otherwise, multiloop PI/PID controllers, which do not incorporate decouplers, could be employed. The decouplers are constructed based on approximate models obtained from proposed biased relay tests. Tuning of the resultant weakly interacting, single-loop PI/PID controllers is then achieved in the predetermined sequence of loop closing. The decouplers, together with single-loop controllers, constitute the multivariable decoupling controller. The proposed design method only assumes that pairing of the manipulated and controlled variables has been made using measures such as the Niederlinski index (Niederlinski, 1971; Grosdidier, et al., 1985) so that the system is PI/PID stabilizable. Simulation results are shown for one 2 × 2 and one 3 × 3 multivariable systems from the literature. The proposed design method compares favorably with the Niederlinski, BLT, and empirical methods. 2. Sequential Tuning of Multiple Control Loops A multivariable decoupling controller consists of n single-loop PID controllers together with n! decouplers. The n × n process GP(s) is represented as follows:
GP(s) ) [gij(s)]
i, j ) 1, 2, ..., n
(1)
The centralized control structure is better illustrated with a 2 × 2 system in Figure 1. Additional transfer function blocks (decouplers) can be introduced between the single-loop controllers and the process. The main objective in decoupling is to compensate for the effect of loop interactions brought about by cross-couplings of the process variables. The design of decouplers will be elaborated in a later section. If the decouplers are absent, the control structure shown in Figure 1 reduces to a multiloop control system. The variables ri(s), ui(s), vi(s), and yi(s) represent, respectively, the reference (setpoint), input, manipulated, and output (controlled) variables of loop i. As depicted in Figure 1, the PID controller of loop i is in
Figure 1. Double-loop decoupling PID control system.
the form of series connection, which consists of the following:
(
Ki(s) ) kCi 1 + Di )
)
1 τIis
1 + τDis 1 + RτDis
(2a)
(2b)
This series connection of the proportional-integral action and the derivative action is utilized to avoid a derivative kick for an abrupt change in the reference input. The term (1 + RτDis) is added to render the controller physically realizable. The value of R is typically between 0.05 and 0.2. Even if the decouplers are incorporated, interactions between the process variables cannot be completely eliminated because of model mismatch. As a result, the multiple control loops for a multivariable process even with decouplers can not, in general, be tuned independently. Here, a sequential tuning strategy that takes interactions into consideration is proposed. The underlying idea is to treat multiple control loops with or without decouplers as a sequence of single loops. For example, the two control loops in the 2 × 2 system can be tuned in a sequential way of loop closing as depicted in Figure 2. Loop 1 is first tuned with loop 2 open (see Figure 2a) by performing a closed-loop identification test on loop 1 to determine the corresponding controller settings. Loop 1 is then placed in automatic with the resultant settings and an identification test is performed on loop 2 (see Figure 2b). With loop 2 closed, an identification test may be performed again on loop 1 to provide a new set of controller settings (see Figure 2c). This sequential tuning procedure can be continued between Figures 2b and 2c until the convergence of all controller parameters is achieved. Note that for each tuning stage, the system will behave like a single loop. The double-loop results can be extended readily to the case of n coupled control loops to be tuned sequentially. It is then clear that multiple control loops can be tuned sequentially in an iterative manner using any single-loop tuning technique. Two questions that arise are: In what sequence should multiple control loops be tuned?; and Why is one sequence advantageous over another? These questions must be answered by examining the mutual effect between the loops via interac-
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Figure 2. Sequential tuning procedure illustrated with a doubleloop control system.
tions. It has been reported several times that in a multiloop control system, a faster loop is less affected via interactions with a slower loop but not vice versa (McAvoy, 1983; Isermann, 1991; Hwang, 1995a). Loh et al. (1993) and Hwang (1995a) further indicated that the speed of a control loop can be estimated roughly by the ultimate frequency of the freely standing loop (i.e., the diagonal element of the process transfer matrix alone). The ultimate frequency is referred to as the frequency of sustained oscillations resulting from a purely proportional control loop. Such considerations result in the rule of thumb that the tuning sequence should be started with the faster loops with higher ultimate frequencies. This sequence allows the slower loops to be tuned later and, hence, enables one to account for the interactions resulting from the closure of the faster loops (Loh et al., 1993). A second rule is that a loop that is significantly faster than some other loops can be treated as a decoupled loop and tuned independently regardless of variations of the controller settings in these slower loops. In this way, the number of iterative tuning steps can be kept to a minimum. Based on the aforementioned arguments and extensive simulation study, a clear and efficient sequential tuning procedure is developed for multiple control loops as follows: Multivariable Decoupling Controller. (1) Without the incorporation of decouplers, perform prior tests on each individual loop (freely standing with all other loops open) to estimate the ultimate frequency. Rank the loop speeds from fast to slow based on the estimated ultimate frequencies.
(2) With the decouplers added, the controller design task is reduced to tuning of several weakly interacting, single-loop controllers, as depicted in Figure 2. The tuning strategy is then to tune the loops once in the fast-to-slow sequence of loop closing. Further iterations are not required because the loop interactions are weak. Multiloop Controllers. (1) Perform prior tests on each individual loop to estimate the ultimate frequency. Rank the loop speeds from fast to slow based on the estimated ultimate frequencies. (2) For a multiloop system, multiple control loops with rather significant interactions may be encountered. The tuning strategy is more elaborate. First, decompose the multiloop system into several subsystems by looking through the estimated ultimate frequencies. Then, let all loops in a more rapid subsystem have ultimate frequencies at least twice those in a slower subsystem. Loops of comparable speeds are grouped into the same subsystem (the ratio of any two adjacent ultimate frequencies is
